© 2011

Spectral Methods

Algorithms, Analysis and Applications


  • The analysis in this book is presented in a unified framework using the non-uniformly weighted Sobolev spaces which lead to simplified analysis and more precise estimates.

  • The book contains, in particular, efficient spectral algorithms and their error analysis for higher-order differential equations, integral equations, problems in unbounded domains and high-dimensional domains.

  • A set of well structured Matlab codes is available online so the readers can easily modify and expand.


Part of the Springer Series in Computational Mathematics book series (SSCM, volume 41)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Jie Shen, Tao Tang, Li-Lian Wang
    Pages 1-22
  3. Jie Shen, Tao Tang, Li-Lian Wang
    Pages 23-46
  4. Jie Shen, Tao Tang, Li-Lian Wang
    Pages 47-140
  5. Jie Shen, Tao Tang, Li-Lian Wang
    Pages 141-180
  6. Jie Shen, Tao Tang, Li-Lian Wang
    Pages 181-200
  7. Jie Shen, Tao Tang, Li-Lian Wang
    Pages 201-236
  8. Jie Shen, Tao Tang, Li-Lian Wang
    Pages 237-298
  9. Jie Shen, Tao Tang, Li-Lian Wang
    Pages 299-366
  10. Jie Shen, Tao Tang, Li-Lian Wang
    Pages 367-413
  11. Back Matter
    Pages 415-470

About this book


Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of  basic convergence theory and error analysis for spectral methods. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online to help the readers to develop their own spectral codes for their specific applications.


numerical analysis orthogonal polynomials/functions scientific computing spectral methods

Authors and affiliations

  1. 1.Dept. MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Dept. MathematicsHong Kong Baptist UniversityKowloonHong Kong/PR China
  3. 3.School of Physical & Mathematical Sci.Nanyang Technological UniversitySingaporeSingapore

About the authors

Jie Shen: Ph.D., Numerical Analysis, Universite de Paris-Sud, Orsay, France, 1987; B.S., Computational Mathematics, Peking University, China, 1982.

Professor of Mathematics at Purdue University; Guest Professorships in Shanghai University and Xiamen University; Member of editorial boards for numerous top research journals.

Tao Tang:  Ph.D., Applied Mathematics, University of Leeds, 1989;
Computational Mathematics, Peking University, China, 1984.

Head and  Chair Professor of Hong Kong Baptist University; Cheung Kong Chair  Professor under Ministry of Education of China; Winner of a Leslie Fox Prize in 1988 and a Feng Kang Prize in Scientific Computing in  2003; Member of editorial boards for numerous  top research journals.

Lilian Wang: Ph.D, Computational Mathematics, Shanghai University, China  2000; B.S., Mathematics Education,  Hunan University of Science and Technology, China, 1995.

Assistant Professor of Mathematics,  Nanyang Technological University, Singapore. A  prolific researcher with over twenty research papers in top journals.

Bibliographic information

Industry Sectors
Oil, Gas & Geosciences
Finance, Business & Banking
Energy, Utilities & Environment


From the reviews:

“This is a largely self-contained book on major parts of the application of spectral methods to the numerical solution of partial differential equations … . The material is accessible to … advanced students of mathematics and also to researchers in neighbouring fields wishing to acquire a sound knowledge of methods they might intend to apply.” (H. Muthsam, Monatshefte für Mathematik, Vol. 170 (2), May, 2013)

“This book provides a self-contained presentation for the construction, implementation and analysis of spectral algorithms for some model equations of elliptic, dispersive and parabolic type. … a textbook for graduate students in mathematics and other sciences and engineering. … The book has nine chapters, each of them ending with a small collection of problems.” (Julia Novo, Mathematical Reviews, January, 2013)

“This is a self-contained presentation on the construction, implementation, and analysis of spectral methods for various differential and integral equations, with wide applications in science and engineering. … Every chapter ends with a set of problems for practice. … This excellent and very well-written book could be used as s graduate textbook in mathematics and other engineering disciplines. It would also be a good reference book for active practitioners and researchers of spectral methods.” (Srinivasan Natesan, ACM Computing Reviews, January, 2013)

“The text consists of nine main chapters, with the material naturally separating into two groups. … the text does a very good job at outlining the importance of creativity and analysis in the development of modern computational techniques for complex applications. … I read the text with enjoyment and expect it to be of great value as a reference, in particular for its careful presentation of key analytic results, and as a comprehensive introduction to the more detailed exposition of the authors’ work … .” (Jan S. Hesthaven, SIAM Review, Vol. 55 (2), 2013)